Program
The Isoperimetric Problem
Student No.:50
Time:【Updated】Mon 13:00-14:50, 2016-03-09~ 2016-05-09(except for May 2)
Instructor:Vlad Moraru  [MSC, Tsinghua University]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2016-3-9
Ending Date:2016-5-9

 

The time of the class will be changed to Monday 13:00-14:50.

 

Description:

 

The course will provide an introduction to several important aspects of the isoperimetric problem, together with some applications to general relativity.

 

Topics will include:

 

1) The isoperimetric inequality in R^2

 

2) The isoperimetric inequality on surfaces with variable curvature

 

3) The isoperimetric inequality on minimal surfaces

 

4) Isoperimetric surfaces and stable constant mean curvature surfaces in Riemannian manifolds

 

5) The isoperimetric profile of a Riemannian manifolds

 

6) Quasi-local mass in general relativity

 

 

Prerequisite:

 

1) A basic course in differential geometry of curves and surfaces (for example chapter 1-3 from do Carmo “Curves and Surfaces”.)

 

2) A basic course in Riemannian geometry (for example, J. M. Lee “Riemannian Geometry”)

 

 

Main References:

 

1) J. Choe, Isoperimetric Inequalities of Minimal Submanifolds (in Global Theory of Minimal Surfaces, Clay Math. Proceedings 2 (2005), 325-369; available at: http://newton.kias.re.kr/~choe/MSRI6.pdf )

 

2) A, Ros, The Isoperimetric Problem (available at: http://www.ugr.es/~aros/isoper.pdf)

 

3) R. Osserman, The isoperimetric inequality Bull. Amer. Math. Soc. Volume 84, Number 6 (1978), 1182-1238. (available at: http://projecteuclid.org/euclid.bams/1183541466)

 

4) J. Barbosa, M. do Carmo, J. Eschenburg, Stability of hypersurfaces of constant mean curvature in Riemannian manifolds Math. Z. , Volume 197,Issue1, pp 123-138 (available at: http://link.springer.com/article/10.1007%2FBF01161634)

Further references will be given during the course.

 

5) M.-T. Wang, Four lectures on quasi-local mass (Preprint available at: http://arxiv.org/abs/1510.02931)